Miscible displacement in bounded fluid layers: Branching beyond critical

Miscible displacement in a bounded fluid layer introduces instabilities due to viscosity variation in the layer and due to viscosity difference across the surfaces separating the layer from surrounding fluid. We find that the problem in the layer itself is similar in many ways to the Bénard problem with the neutral curve exhibiting the characteristic dip leading to the possibility of many flow cells at the critical speed of displacement.The viscosity difference across the surface modifies the dip, slightly at large surface tensions, completely at low surface tensions. The effect of the surface deflection is present whether or not the surface is stable and can be omitted only if the viscosity of the driving fluid is very high or the surface tension is very low.The nature of the branching at the critical speed depends on the cross section of the fluid layer and on how much of the dip in the neutral curve remains.